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THE PROPER AND SEMI-PROPER FORCING AXIOMS FOR FORCING NOTIONS THAT PRESERVE ℵ2 OR ℵ3

By Joel David Hamkins and Thomas A. Johnstone

Abstract

We prove that the PFA lottery preparation of a strongly unfoldable cardinal κ under ¬0 ♯ forces PFA(ℵ2-preserving), PFA(ℵ3-preserving) and PFAℵ2,with2ω = κ = ℵ2. The method adapts to semi-proper forcing, giving SPFA(ℵ2-preserving), SPFA(ℵ3-preserving) and SPFAℵ2 from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction SPFA(ℵ2-preserving)+ SPFA(ℵ3-preserving) + SPFAℵ2 +2ω = ℵ2. Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse ℵ3 to ℵ1

Year: 2008
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.4804
Provided by: CiteSeerX
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